Each linear minion corresponds to a relaxation for (P)CSPs, through the notion of free structure.We define the free structure for uniform hypergraphs rather than digraphs, because we will later use it in that more general case. In fact, the same construction can be applied to arbitrary relational structures, see [1]}. Intuitively, the free structure of a linear minion \({{M}}\)  generated by a hypergraph \(\mathbf {H}\)  simulates the structure of \(\mathbf {H}\)  inside \({{M}}\) : The vertices become matrices of \({{M}}\) , while the hyperedges are tuples of matrices that can all be obtained from a single other matrix through elementary row operations.
The formal definition
is as follows:
